Question

By what smallest number $3645$ be multiplied so that the product becomes a perfect cube?

Answer 1

Prime factorising $3645$, we get,

$3645=5\times 3\times 3\times 3\times 3\times 3\times 3$

$=5^1\times 3^6$.

We know, a perfect cube has multiples of $3$ as powers of prime factors.

Here, number of $5$'s is $1$ and number of $3$'s is $6$.

So we need to multiply another $5^2$ to the factorization to make $3645$ a perfect cube.

Hence, the smallest number by which $3645$ must be multiplied to obtain a perfect cube is $5^2=25$.